Microfoundations, the New Keynesian Model, and Solution Methods

What is a DSGE Model?

Dynamic Stochastic General Equilibrium models are the workhorse of modern macroeconomics. They combine:

Component Meaning
Dynamic Agents optimize over time, considering future
Stochastic Random shocks drive business cycle fluctuations
General All markets (goods, labor, capital) interact
Equilibrium Supply equals demand in every market
NoteThe Core Philosophy

DSGE models are microfounded: aggregate behavior emerges from explicit optimization by households, firms, and policymakers. No ad hoc aggregate relationships.

From Micro to Macro

DSGE model structure

The General Form

All DSGE models can be written as:

\[ \mathbb{E}_t\left[f(y_{t+1}, y_t, y_{t-1}, u_t)\right] = 0 \]

where:

  • \(y_t\) = vector of endogenous variables (output, inflation, interest rate, capital, …)
  • \(u_t\) = vector of exogenous shocks (technology, monetary, fiscal, …)
  • \(f\) = system of equilibrium conditions (Euler equations, market clearing, …)
  • \(\mathbb{E}_t\) = expectation conditional on time-\(t\) information

The challenge: solving for \(y_t\) as a function of states and shocks.

The RBC Core

Before New Keynesian elements, we start with the Real Business Cycle foundation.

Household Problem

A representative household maximizes expected lifetime utility:

\[ \max_{\{C_t, N_t, K_t\}} \mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t U(C_t, N_t) \]

Subject to budget constraint: \[ C_t + K_t = W_t N_t + R_t^K K_{t-1} + \Pi_t - T_t + (1-\delta)K_{t-1} \]

where: - \(C_t\) = consumption - \(N_t\) = labor supply - \(K_t\) = end-of-period capital - \(W_t\) = real wage - \(R_t^K\) = rental rate on capital - \(\delta\) = depreciation rate - \(\Pi_t\) = firm profits - \(T_t\) = lump-sum taxes

First-Order Conditions

Euler equation (intertemporal consumption choice): \[ U_C(C_t, N_t) = \beta \mathbb{E}_t\left[U_C(C_{t+1}, N_{t+1}) \cdot (R_{t+1}^K + 1 - \delta)\right] \]

Labor supply (consumption-leisure trade-off): \[ \frac{-U_N(C_t, N_t)}{U_C(C_t, N_t)} = W_t \]

Example: Log Utility

With \(U(C, N) = \log(C) - \chi \frac{N^{1+\phi}}{1+\phi}\):

\[ \frac{1}{C_t} = \beta \mathbb{E}_t\left[\frac{1}{C_{t+1}} (R_{t+1}^K + 1 - \delta)\right] \]

\[ \chi N_t^\phi \cdot C_t = W_t \]

Code 
# Illustrate consumption smoothing
beta <- 0.99
R_gross <- 1.02  # gross interest rate
sigma <- 1  # CRRA coefficient

# For sigma = 1 (log utility): C_t+1/C_t = (beta * R)
growth_rate <- beta * R_gross

# Simulate consumption path with and without smoothing
T_sim <- 20
income <- c(rep(1, 10), rep(1.2, 10))  # income jumps at t=10

# Without smoothing (hand-to-mouth)
c_htm <- income

# With smoothing (optimal)
# Permanent income = average
perm_inc <- mean(income)
c_smooth <- rep(perm_inc, T_sim)

consumption_df <- tibble(
  t = rep(1:T_sim, 2),
  consumption = c(c_htm, c_smooth),
  type = rep(c("Hand-to-mouth", "Euler optimal"), each = T_sim)
)

ggplot(consumption_df, aes(x = t, y = consumption, color = type, linetype = type)) +
  geom_line(linewidth = 1.2) +
  geom_vline(xintercept = 10, linetype = "dashed", alpha = 0.5) +
  annotate("text", x = 10.5, y = 1.15, label = "Income jump", hjust = 0) +
  scale_color_manual(values = c("#e74c3c", "#3498db")) +
  labs(title = "Consumption Smoothing via the Euler Equation",
       subtitle = "Optimal: smooth consumption; Suboptimal: track income",
       x = "Period", y = "Consumption", color = NULL, linetype = NULL) +
  theme_minimal() +
  theme(legend.position = "bottom")

Euler equation: optimal consumption smoothing

Firm Problem

A representative firm maximizes profits with Cobb-Douglas production:

\[ Y_t = A_t K_{t-1}^\alpha N_t^{1-\alpha} \]

Factor demands (competitive markets, firms are price-takers): \[ W_t = (1-\alpha) \frac{Y_t}{N_t} \] \[ R_t^K = \alpha \frac{Y_t}{K_{t-1}} \]

Technology shock follows AR(1): \[ \log(A_t) = \rho_A \log(A_{t-1}) + \varepsilon_{A,t}, \quad \varepsilon_{A,t} \sim N(0, \sigma_A^2) \]

Market Clearing

\[ Y_t = C_t + I_t \]

where investment \(I_t = K_t - (1-\delta)K_{t-1}\).

The Complete RBC Model

Equation Name Variables
\(\frac{1}{C_t} = \beta \mathbb{E}_t\left[\frac{1}{C_{t+1}}(R_{t+1}^K + 1 - \delta)\right]\) Euler \(C, R^K\)
\(\chi N_t^\phi C_t = W_t\) Labor supply \(N, C, W\)
\(Y_t = A_t K_{t-1}^\alpha N_t^{1-\alpha}\) Production \(Y, A, K, N\)
\(W_t = (1-\alpha) Y_t / N_t\) Labor demand \(W, Y, N\)
\(R_t^K = \alpha Y_t / K_{t-1}\) Capital demand \(R^K, Y, K\)
\(Y_t = C_t + K_t - (1-\delta)K_{t-1}\) Resource \(Y, C, K\)
\(\log A_t = \rho_A \log A_{t-1} + \varepsilon_A\) Technology \(A\)

7 equations, 7 unknowns: \((Y, C, N, K, W, R^K, A)\)

The New Keynesian Model

The RBC model has no role for monetary policy (all real). The New Keynesian model adds:

  1. Nominal rigidities (sticky prices/wages)
  2. Monopolistic competition (firms have pricing power)
  3. Monetary policy (central bank sets interest rate)

The 3-Equation NK Model

The canonical log-linearized NK model:

1. IS Curve (Dynamic IS)

From the household’s Euler equation: \[ \hat{y}_t = \mathbb{E}_t[\hat{y}_{t+1}] - \frac{1}{\sigma}(i_t - \mathbb{E}_t[\pi_{t+1}] - r_t^n) \]

where: - \(\hat{y}_t\) = output gap (deviation from flexible-price equilibrium) - \(i_t\) = nominal interest rate - \(\pi_t\) = inflation - \(r_t^n\) = natural rate of interest - \(\sigma\) = inverse elasticity of intertemporal substitution

Interpretation: Higher real interest rate → postpone consumption → lower output today.

2. Phillips Curve (NKPC)

From Calvo pricing (fraction \(1-\theta\) of firms adjust each period): \[ \pi_t = \beta \mathbb{E}_t[\pi_{t+1}] + \kappa \hat{y}_t \]

where: \[ \kappa = \frac{(1-\theta)(1-\beta\theta)}{\theta} \cdot (\sigma + \phi) \]

Interpretation: Inflation today depends on expected future inflation plus current output gap (marginal cost pressure).

3. Taylor Rule

Central bank sets interest rate: \[ i_t = \rho_i i_{t-1} + (1-\rho_i)\left[r^* + \phi_\pi(\pi_t - \pi^*) + \phi_y \hat{y}_t\right] + \varepsilon_{m,t} \]

Standard calibration:

Parameter Value Interpretation
\(\rho_i\) 0.8 Interest rate smoothing
\(\phi_\pi\) 1.5 Response to inflation (Taylor principle: \(\phi_\pi > 1\))
\(\phi_y\) 0.5/4 = 0.125 Response to output gap (quarterly)
Code 
# Solve simple 3-eq NK model
# Parameters
beta <- 0.99
sigma <- 1
kappa <- 0.1
phi_pi <- 1.5
phi_y <- 0.125
rho_i <- 0.8

# Monetary policy shock simulation
T_sim <- 40
eps_m <- c(0, 0.25, rep(0, T_sim - 2))  # 25bp shock at t=2

# Initialize
y_gap <- pi <- i_rate <- numeric(T_sim)
y_gap[1] <- pi[1] <- i_rate[1] <- 0

# Simple forward-looking solution (assuming expectations = 0 for simplicity)
# More rigorous: use Blanchard-Kahn
for (t in 2:T_sim) {
  # Taylor rule
  i_rate[t] <- rho_i * i_rate[t-1] + (1 - rho_i) * (phi_pi * pi[t-1] + phi_y * y_gap[t-1]) + eps_m[t]

  # IS curve (simplified: E[y+1] = 0, E[pi+1] = 0)
  y_gap[t] <- -1/sigma * (i_rate[t] - 0 - 0) + 0.7 * y_gap[t-1]

  # Phillips curve
  pi[t] <- beta * 0 + kappa * y_gap[t] + 0.5 * pi[t-1]
}

nk_df <- tibble(
  t = rep(1:T_sim, 3),
  value = c(y_gap * 100, pi * 100, i_rate * 100),
  variable = rep(c("Output Gap (%)", "Inflation (%)", "Interest Rate (%)"), each = T_sim)
)

ggplot(nk_df, aes(x = t, y = value, color = variable)) +
  geom_line(linewidth = 1) +
  geom_hline(yintercept = 0, linetype = "dashed", alpha = 0.5) +
  facet_wrap(~variable, scales = "free_y", ncol = 1) +
  labs(title = "Response to 25bp Monetary Policy Shock",
       subtitle = "Simplified 3-equation NK model",
       x = "Quarters", y = "Percent deviation") +
  scale_color_manual(values = c("#3498db", "#e74c3c", "#2ecc71")) +
  theme_minimal() +
  theme(legend.position = "none")

The 3-equation New Keynesian model

Deriving the Phillips Curve

Calvo Pricing Setup

  • Each period, fraction \(1-\theta\) of firms can reset prices
  • Fraction \(\theta\) are “stuck” with last period’s price
  • When firms can adjust, they set price to maximize expected profits over the time they’ll be stuck

Optimal Price Setting

A firm that can adjust at time \(t\) chooses \(P_t^*\) to maximize: \[ \mathbb{E}_t \sum_{k=0}^{\infty} (\beta\theta)^k \left[ \frac{P_t^*}{P_{t+k}} Y_{t+k|t} - MC_{t+k} Y_{t+k|t} \right] \]

Result (after log-linearization): \[ \hat{p}_t^* = (1-\beta\theta) \sum_{k=0}^{\infty} (\beta\theta)^k \mathbb{E}_t[\widehat{mc}_{t+k}] \]

Firms set prices as a weighted average of expected future marginal costs.

Aggregating to the Phillips Curve

Price index evolution: \[ P_t = \left[\theta P_{t-1}^{1-\epsilon} + (1-\theta)(P_t^*)^{1-\epsilon}\right]^{\frac{1}{1-\epsilon}} \]

Log-linearizing and combining: \[ \pi_t = \beta \mathbb{E}_t[\pi_{t+1}] + \kappa \cdot \widehat{mc}_t \]

With \(\widehat{mc}_t \propto \hat{y}_t\) (output gap approximates marginal cost deviations).

Log-Linearization

DSGE models are nonlinear. To solve them analytically (or with first-order perturbation), we log-linearize around the steady state.

The Technique

For any variable \(X_t\):

\[ \hat{x}_t \equiv \log(X_t) - \log(\bar{X}) = \log\left(\frac{X_t}{\bar{X}}\right) \approx \frac{X_t - \bar{X}}{\bar{X}} \]

So \(\hat{x}_t\) is the percentage deviation from steady state.

Key Approximation

For any function \(f(X_t, Y_t)\): \[ f(X_t, Y_t) \approx f(\bar{X}, \bar{Y}) + f_X(\bar{X}, \bar{Y})(X_t - \bar{X}) + f_Y(\bar{X}, \bar{Y})(Y_t - \bar{Y}) \]

Example: Euler Equation

Nonlinear Euler: \[ \frac{1}{C_t} = \beta \mathbb{E}_t\left[\frac{1}{C_{t+1}} R_{t+1}\right] \]

Step 1: Steady state \[ \frac{1}{\bar{C}} = \beta \frac{1}{\bar{C}} \bar{R} \implies \bar{R} = \frac{1}{\beta} \]

Step 2: Log-linearize. Let \(C_t = \bar{C} e^{\hat{c}_t} \approx \bar{C}(1 + \hat{c}_t)\): \[ \frac{1}{\bar{C}(1 + \hat{c}_t)} = \beta \mathbb{E}_t\left[\frac{1}{\bar{C}(1 + \hat{c}_{t+1})} \bar{R}(1 + \hat{r}_{t+1})\right] \]

Step 3: First-order Taylor expansion (ignore second-order terms): \[ 1 - \hat{c}_t = \beta \bar{R} \mathbb{E}_t\left[(1 - \hat{c}_{t+1})(1 + \hat{r}_{t+1})\right] \] \[ 1 - \hat{c}_t \approx \mathbb{E}_t[1 - \hat{c}_{t+1} + \hat{r}_{t+1}] \]

Result: \[ \hat{c}_t = \mathbb{E}_t[\hat{c}_{t+1}] - (\hat{r}_{t+1} - 0) \]

This is the log-linearized Euler equation (with \(\sigma = 1\)).

Code 
# Compare nonlinear vs log-linear Euler
C_ss <- 1
beta <- 0.99
R_ss <- 1/beta

# Deviations from steady state
c_dev <- seq(-0.2, 0.2, 0.01)

# Nonlinear: U'(C) = 1/C
uprime_nonlinear <- 1 / (C_ss * (1 + c_dev))

# Log-linear: U'(C) ≈ 1/C_ss * (1 - c_dev)
uprime_linear <- (1/C_ss) * (1 - c_dev)

approx_df <- tibble(
  deviation = rep(c_dev * 100, 2),
  marginal_utility = c(uprime_nonlinear, uprime_linear),
  method = rep(c("Nonlinear", "Log-linear"), each = length(c_dev))
)

ggplot(approx_df, aes(x = deviation, y = marginal_utility, color = method)) +
  geom_line(linewidth = 1.2) +
  geom_vline(xintercept = 0, linetype = "dashed", alpha = 0.5) +
  labs(title = "Log-Linearization Accuracy",
       subtitle = "Approximation works well for small deviations (±10%)",
       x = "% Deviation from Steady State",
       y = "Marginal Utility U'(C)", color = NULL) +
  scale_color_manual(values = c("#3498db", "#e74c3c")) +
  theme_minimal() +
  theme(legend.position = "bottom")

Log-linearization accuracy

Solving Rational Expectations Models

After log-linearization, DSGE models take the form:

\[ A \mathbb{E}_t[y_{t+1}] = B y_t + C u_t \]

where \(y_t\) contains both predetermined (state) and jump (control) variables.

Partitioning the System

Split \(y_t = \begin{pmatrix} x_t \\ z_t \end{pmatrix}\) where:

  • \(x_t\): Predetermined (state) variables — known at time \(t\) (e.g., capital \(K_{t-1}\))
  • \(z_t\): Jump (forward-looking) variables — can adjust freely (e.g., consumption, inflation)

Blanchard-Kahn Conditions

The eigenvalue decomposition of the system matrix determines solvability (Blanchard and Kahn 1980).

ImportantBlanchard-Kahn (1980) Conditions

For a unique stable solution:

Number of eigenvalues outside unit circle = Number of jump variables

Eigenvalue Count Diagnosis
Equals number of jump vars Unique solution (saddle-path stable)
Less than number of jump vars Indeterminacy (multiple equilibria, sunspots)
Greater than number of jump vars No solution (explosive dynamics)

The Solution Form

If BK conditions hold, the solution is:

\[ x_{t+1} = H_x x_t + H_u u_{t+1} \] \[ z_t = G_x x_t \]

The policy function \(G_x\) tells us how jump variables respond to states. The transition matrix \(H_x\) governs state evolution.

Code 
# Illustrate saddle-path in a simple 2-variable system
# x_t = state (capital), z_t = control (consumption)

# Phase diagram for consumption-capital
k_grid <- seq(0.5, 1.5, 0.02)
c_grid <- seq(0.5, 1.5, 0.02)

# Steady state (normalized)
k_ss <- c_ss <- 1

# Simplified dynamics for illustration
# dk/dt = 0 locus: c = f(k) - delta*k
# dc/dt = 0 locus: f'(k) = rho + delta

alpha <- 0.33
delta <- 0.025
rho <- 0.01

# Nullclines
c_from_k_nullcline <- function(k) k^alpha - delta * k
k_from_c_nullcline <- (alpha / (rho + delta))^(1/(1-alpha))

# Create phase diagram
phase_df <- expand_grid(k = k_grid, c = c_grid) %>%
  mutate(
    dk = k^alpha - c - delta * k,
    dc = (alpha * k^(alpha-1) - delta - rho) * c / 2  # simplified
  )

# Saddle path (approximate)
saddle_path <- tibble(
  k = seq(0.7, 1.3, 0.01),
  c = c_ss + 0.8 * (k - k_ss)  # slope of stable eigenvector
)

ggplot() +
  # Direction field
  geom_segment(data = phase_df %>% sample_n(200),
               aes(x = k, y = c, xend = k + dk*0.05, yend = c + dc*0.05),
               arrow = arrow(length = unit(0.1, "cm")), alpha = 0.3) +
  # Nullclines
  geom_function(fun = c_from_k_nullcline, color = "#3498db", linewidth = 1.2) +
  geom_vline(xintercept = k_from_c_nullcline, color = "#e74c3c", linewidth = 1.2) +
  # Saddle path
  geom_line(data = saddle_path, aes(x = k, y = c),
            color = "#2ecc71", linewidth = 1.5, linetype = "dashed") +
  # Steady state
  geom_point(aes(x = k_ss, y = c_ss), size = 4, color = "black") +
  annotate("text", x = k_ss + 0.05, y = c_ss + 0.05, label = "Steady State") +
  annotate("text", x = 1.3, y = c_from_k_nullcline(1.3) - 0.05,
           label = "dk = 0", color = "#3498db") +
  annotate("text", x = k_from_c_nullcline + 0.05, y = 1.4,
           label = "dc = 0", color = "#e74c3c") +
  annotate("text", x = 1.2, y = c_ss + 0.8 * 0.2 + 0.05,
           label = "Saddle Path", color = "#2ecc71") +
  labs(title = "Saddle-Path Stability in DSGE",
       subtitle = "Jump variable (c) must be on saddle path for convergence",
       x = "Capital (k)", y = "Consumption (c)") +
  coord_cartesian(xlim = c(0.6, 1.4), ylim = c(0.6, 1.4)) +
  theme_minimal()

Blanchard-Kahn: saddle-path stability

Common Causes of BK Violations

Problem Symptom Common Fix
Taylor principle violated Too few unstable roots Raise \(\phi_\pi\) above 1
Missing expectations Wrong root count Check all \(\mathbb{E}_t\) terms
Timing error Explosive solutions Dynare uses end-of-period capital: use \(K(-1)\)
Wrong variable classification Indeterminacy Re-check predetermined vs. jump

Numerical Solution: QZ Decomposition

For larger models, we use the generalized Schur (QZ) decomposition:

\[ A = Q \Lambda Z', \quad B = Q \Omega Z' \]

where \(\Lambda/\Omega\) gives generalized eigenvalues. Reorder to put unstable eigenvalues in the bottom block.

# R code sketch for BK solution
solve_dsge_bk <- function(A, B, n_state) {
  # A * E[y_{t+1}] = B * y_t
  # y = [x; z] where x = states, z = jumps

  n_total <- nrow(A)
  n_jump <- n_total - n_state

  # QZ decomposition
  qz <- geigen::gqz(A, B)  # Generalized Schur

  # Eigenvalues
  eig <- qz$alpha / qz$beta
  n_unstable <- sum(abs(eig) > 1)

  # Check BK conditions
  if (n_unstable != n_jump) {
    stop(paste("BK violation:", n_unstable, "unstable,", n_jump, "jump vars"))
  }

  # Extract policy function (from stable block)
  # ... (reorder and partition)

  list(G = G_matrix, H = H_matrix, eigenvalues = eig)
}

Calibration

Before estimation (Module 11), we calibrate parameters using:

  1. Microeconomic evidence (labor supply elasticity, depreciation rate)
  2. Long-run averages (capital-output ratio, hours worked)
  3. Steady-state relationships (Euler equation pins down \(\beta\))

Standard RBC/NK Calibration

Parameter Symbol Value Source
Discount factor \(\beta\) 0.99 Real interest rate ≈ 4% annually
Capital share \(\alpha\) 0.33 National accounts
Depreciation \(\delta\) 0.025 10% annual depreciation
Risk aversion \(\sigma\) 1-2 Micro studies
Frisch elasticity \(1/\phi\) 0.5-2 Labor supply literature
Calvo parameter \(\theta\) 0.75 Prices adjust every 4 quarters
Inflation weight \(\phi_\pi\) 1.5 Taylor (1993)
Output weight \(\phi_y\) 0.125 Taylor (1993), quarterly

Matching Moments

Target: Model-implied moments ≈ Data moments

Moment Data (US) Typical RBC
std(Y) 1.5-2% Match by calibrating \(\sigma_A\)
std(C)/std(Y) 0.5-0.7 Endogenous
std(I)/std(Y) 2.5-3.5 Endogenous
corr(C, Y) 0.8-0.9 Endogenous
corr(N, Y) 0.8-0.9 Endogenous
Code 
# Simulate RBC model and compute moments
# Simplified AR(1) approximation

# Parameters
rho_a <- 0.95
sigma_a <- 0.007
T_sim <- 200

# Technology shock
set.seed(123)
log_a <- numeric(T_sim)
log_a[1] <- 0
for (t in 2:T_sim) {
  log_a[t] <- rho_a * log_a[t-1] + rnorm(1, 0, sigma_a)
}

# Simplified responses (from RBC solution)
# Output: y_hat ≈ (1 + alpha/(1-alpha)) * a_hat
# Consumption: c_hat ≈ 0.7 * y_hat (consumption smoother than output)
# Investment: i_hat ≈ 3 * y_hat (investment more volatile)
# Hours: n_hat ≈ 0.5 * y_hat

y_hat <- 1.5 * log_a
c_hat <- 0.7 * y_hat + rnorm(T_sim, 0, 0.002)
i_hat <- 3.0 * y_hat + rnorm(T_sim, 0, 0.005)
n_hat <- 0.5 * y_hat + rnorm(T_sim, 0, 0.003)

# Compute moments
moments_model <- tibble(
  Variable = c("Output", "Consumption", "Investment", "Hours"),
  `Std Dev (%)` = c(sd(y_hat), sd(c_hat), sd(i_hat), sd(n_hat)) * 100,
  `Rel to Y` = c(1, sd(c_hat)/sd(y_hat), sd(i_hat)/sd(y_hat), sd(n_hat)/sd(y_hat)),
  `Corr with Y` = c(1, cor(c_hat, y_hat), cor(i_hat, y_hat), cor(n_hat, y_hat))
)

# Add data targets
moments_data <- tibble(
  Variable = c("Output", "Consumption", "Investment", "Hours"),
  `Std Dev (%)` = c(1.8, 1.0, 5.0, 1.5),
  `Rel to Y` = c(1, 0.55, 2.8, 0.83),
  `Corr with Y` = c(1, 0.85, 0.90, 0.85)
)

# Display
knitr::kable(
  bind_rows(
    moments_model %>% mutate(Source = "Model"),
    moments_data %>% mutate(Source = "Data")
  ) %>%
    pivot_wider(names_from = Source, values_from = c(`Std Dev (%)`, `Rel to Y`, `Corr with Y`)),
  caption = "Business Cycle Moments: Model vs. Data",
  digits = 2
)
Business Cycle Moments: Model vs. Data
Variable Std Dev (%)_Model Std Dev (%)_Data Rel to Y_Model Rel to Y_Data Corr with Y_Model Corr with Y_Data
Output 2.26 1.8 1.00 1.00 1.00 1.00
Consumption 1.60 1.0 0.71 0.55 0.99 0.85
Investment 6.77 5.0 2.99 2.80 1.00 0.90
Hours 1.14 1.5 0.50 0.83 0.96 0.85

Calibration: matching business cycle moments

Steady-State Relationships

Many parameters are pinned down by steady-state conditions:

From Euler equation: \[ \bar{R} = \frac{1}{\beta} \implies \beta = \frac{1}{1 + r^*} \] With \(r^* = 4\%\) annually → \(\beta = 1/1.01 \approx 0.99\) quarterly.

From capital demand: \[ \bar{R}^K = \alpha \frac{\bar{Y}}{\bar{K}} = \frac{1}{\beta} - (1-\delta) \] Given \(\beta\), \(\delta\), and capital-output ratio, backs out \(\alpha\).

From labor supply: \[ \chi \bar{N}^\phi \bar{C} = \bar{W} = (1-\alpha)\frac{\bar{Y}}{\bar{N}} \] Given target \(\bar{N} = 1/3\) (8 hours/day), backs out \(\chi\).

Dynare Basics

Dynare is the standard software for solving and estimating DSGE models. Here’s a minimal example.

A Simple RBC Model in Dynare

%% RBC Model - rbc_simple.mod
%% Preamble
var y c k n w r a;
varexo eps_a;
parameters BETA ALPHA DELTA RHO_A SIGMA_A CHI PHI;

%% Calibration
BETA = 0.99;
ALPHA = 0.33;
DELTA = 0.025;
RHO_A = 0.95;
SIGMA_A = 0.007;
CHI = 1;       % labor disutility
PHI = 1;       % inverse Frisch elasticity

%% Model equations
model;
  % Euler equation
  1/c = BETA * (1/c(+1)) * (r(+1) + 1 - DELTA);

  % Labor supply
  CHI * n^PHI * c = w;

  % Production function
  y = a * k(-1)^ALPHA * n^(1-ALPHA);

  % Factor prices
  w = (1-ALPHA) * y / n;
  r = ALPHA * y / k(-1);

  % Resource constraint
  y = c + k - (1-DELTA)*k(-1);

  % Technology shock
  log(a) = RHO_A * log(a(-1)) + eps_a;
end;

%% Steady state
initval;
  a = 1;
  r = 1/BETA - 1 + DELTA;
  k = (ALPHA/r)^(1/(1-ALPHA)) * 0.33;
  n = 0.33;
  y = a * k^ALPHA * n^(1-ALPHA);
  c = y - DELTA*k;
  w = (1-ALPHA) * y / n;
end;

steady;
check;

%% Shocks
shocks;
  var eps_a; stderr SIGMA_A;
end;

%% Simulation
stoch_simul(order=1, irf=40, periods=200);

Key Dynare Commands

Command Purpose
steady Compute deterministic steady state
check Verify Blanchard-Kahn conditions
stoch_simul(order=1) First-order perturbation solution
stoch_simul(irf=40) Generate IRFs for 40 periods
estimation(...) Bayesian estimation (Module 11)
model_diagnostics Check model specification

Important Timing Convention

WarningDynare Timing

Dynare uses end-of-period capital: - k without subscript = \(K_t\) (end of period \(t\)) - k(-1) = \(K_{t-1}\) (beginning of period \(t\) = end of \(t-1\))

So the production function uses k(-1):

y = a * k(-1)^ALPHA * n^(1-ALPHA);

Summary

Concept Key Insight
DSGE structure Microfounded optimization → aggregate equilibrium
Households Euler equation governs intertemporal choice
Firms (NK) Calvo pricing → forward-looking Phillips curve
Monetary policy Taylor rule with \(\phi_\pi > 1\) for stability
Log-linearization Approximate around steady state for solution
Blanchard-Kahn Saddle-path stability requires correct eigenvalue count
Calibration Match parameters to micro evidence and steady-state targets
TipNext Steps

Module 11 covers DSGE estimation: - Bayesian methods with Metropolis-Hastings - The Kalman filter for likelihood evaluation - Model comparison via marginal likelihood - Dynare’s estimation command

Key References

Foundational

  • Kydland & Prescott (1982) “Time to Build and Aggregate Fluctuations” Econometrica — RBC origins
  • Blanchard & Kahn (1980) “The Solution of Linear Difference Models” Econometrica — BK conditions
  • Galí (2015) Monetary Policy, Inflation, and the Business Cycle — NK textbook

Medium-Scale Models

  • Smets & Wouters (2003, 2007) “An Estimated DSGE Model” — Benchmark estimated NK
  • Christiano, Eichenbaum & Evans (2005) “Nominal Rigidities” JPE — CEE model

Solution Methods

  • Uhlig (1999) “A Toolkit for Analysing Nonlinear Dynamic Stochastic Models” — Perturbation
  • Sims (2002) “Solving Linear Rational Expectations Models” — QZ decomposition

Calibration

  • Cooley & Prescott (1995) “Economic Growth and Business Cycles” — Calibration methodology
  • Chari, Kehoe & McGrattan (2007) “Business Cycle Accounting” Econometrica — Wedges approach

Software

  • Adjemian et al. “Dynare” — Standard DSGE solver
  • Herbst & Schorfheide (2016) Bayesian Estimation of DSGE Models — Modern estimation